[Badiou and Science] 1.2 Infinity and Infinite Sets

“… the existence of zero, or the empty set, and the existence of an infinite set can in no way be deduced from “purely logical” presuppositions. They are axiomatic decisions, taken under the constraints of the historical injunction of being… The two axioms of the void and of the infinite structure the entire thinking of number… The fact that this is a matter of axioms and not of theorems means that the existence of zero and of the infinite is prescribed to thought by being…” -Alain Badiou, Number and Numbers, pg. 56–57

To review, we have already introduced some of the axioms of set theory as we know them today. In terms of their historical development we’re still a ways off from their eventual formalization, but here is what we have so far:

Axiom of the Empty Set: there exists a set with no elements.
- This asserts the existence of “zero”. Badiou also refers to this set as the “void”. So empty set == void == zero.

Axiom of the Power Set: given any set, there is the set composed of all of its parts or inclusions.
- As previously elucidated, for a set of size n its power set will have a size of 2ⁿ.

With just the concept of a set, the primal relation of “belonging”, the two axioms above, and a rule of succession (which will also be axiomatized) we can construct all of the ordinals (= natural numbers = positive integers = 1, 2, 3, etc.). Impressive stuff, right? We will also recognize that basic operations of arithmetic, such as addition and multiplication, can be defined over these sets with operators built off these basic notions. Though not important to our project here, interested readers can further research how this is accomplished to convince themselves that the theory so far is adequate to found even our basic mathematics and arithmetic.

The Counting Numbers and Succession
The power set acts as a function of “succession” for the ordinals. Given an ordinal W, “W+1” is equal to the power set of W. So, every ordinal is a successor ordinal (that is, every whole number succeeds another). Interestingly, the ordinal that is succeeded by “W+1” is actually the “maximum element” of “W+1”, which should be W.

For example, since 5 = [0 1 2 3 4] its maximum element is 4, which is the very ordinal that 5 succeeds.

All ordinals are successors and have a maximum element. Such a property is constitutive of being an ordinal number. However, as one can see, this is patently not true of the empty set — it neither succeeds nor has a maximum element (it, in fact, has no elements). Therefore the empty set’s existence must be secured via our Axiom of the Empty Set, and 0 is not strictly considered an ordinal. However, note that the empty set which 0 represents is still always an implicit subset of every set.

The Limit Ordinal and Infinity
The genius of Cantor was to then define a “limit ordinal”, ωₒ, which represents the set of all these possible ordinal numbers. The guiding idea is that the limit ordinal is a “special” ordinal that does not succeed another ordinal and consequently does not have a maximum element. The limit ordinal represents the ordinal “greater than” every other ordinal, for which between it and any ordinal contained within it, there must be another ordinal (an “infinity” of ordinals in the common parlance).

Therefore, all ordinal numbers belong to the set ωₒ (they are all contained within the set ωₒ) and there are always successor ordinals for any given ordinal within this set. As such, ωₒ is the “limit” of this set and is the first “infinite” ordinal proper. Very simply put:

ωₒ= [0 1 2 3 4…]

ωₒ is thus knighted the first “infinity”, and it contains all ordinals.

Size and Cardinality
One may then ask, what is the “size” of ωₒ? When we have spoken of the size of finite sets, they have simply been assigned a size equal to their “natural number” of elements. So, the set [0 1 2 3 4] has a size of 5, and its power set has a size 2⁵ = 32. The set [Eddie Alex David Michael] has size 4, and its power set has size 2⁴ = 16. The general term for the “size” of a set is its “cardinality”. When describing finite sets, like these latter examples, this is straightforward.

But what about this new infinite set ωₒ? We assign this set the cardinality of אₒ; this being the first “infinite” or “transfinite” cardinal [1]. So while all finite sets are “numbered” or “counted” by how many elements they contain via the finite ordinals (e.g. 1, 2, 3…), infinite sets require a new counting system using the infinite cardinals. The cardinality אₒ represents the “number” of elements contained in the first infinite set ωₒ. So ωₒ is said to have אₒ elements — an “infinity” of elements.

It suffices to describe ωₒ as the first set which, while “full” of all the ordinal numbers, does not succeed from a previous number (there is no “infinity” prior to this first infinity). Just as how the empty set forms the initial basis for construction of the finite ordinals with no predecessor of its own, ωₒ is the basis for infinite numbers and אₒ the basis for infinite cardinals:

The limit ordinal ωₒ is announced by the Axiom of Infinity: there is a set that contains the empty set, and is such that if a belongs to it then so does the power set of a = p(a).

Deciding the Infinite
One quickly notices that we have left the stable ground of empirical science when talking of the infinite and of infinities — there is no hope of observing any of these entities in the world. The existence of such concepts requires a pure decision in the form of an axiom. Badiou observes,

“On condition of the existence of the void [empty set], there is 1, and 2, and 3…, all successors. But a limit ordinal? …we find ourselves on the verge of the decision on the infinite. No hope of [empirically] proving the existence of a single limit ordinal. We must make the great modern declaration: the infinite exists, and, what is more, it exists in a wholly banal sense, being neither revealed (religion), nor proved (mediaeval metaphysics), but being simply decided, under the injunction of being, in the form of number… That is infinite which, not being void [or empty], meanwhile does not succeed.” [Number and Numbers, pg. 82]

Asserting the existence of the limit ordinal represents an “unleashing of thought”. A pure decision. We have left any sort of empirical territory (certain to cause some unease and nausea amongst many of us scientists), but we have been catapulted to a space where we must rely on pure thought to work out the consequences of these concepts and the possibilities they hold. As always, we could stubbornly deny the axiomatic existence of the empty set and the infinite (or worse, relegate it to the “mystical”), but it would unravel our whole theory, which has been pretty consistent so far, and rend it asunder. Therefore, we must brazenly venture on and explore the uncertain…

We now have a theory that defines all of our basic numbers and prescribes their endless limit in a concept of infinity. But what about our other familiar numbers, such as fractions and irrational numbers (like π)? Are they contained within this theory?

Stay tuned, and as always sound off in the comments!

- Dr. G

Next: The Continuum Hypothesis

Notes

[1] For Cantor’s specific construction of ωₒ and אₒ, see: Mary Tiles, The Philosophy of Set Theory, 104–107 and/or Howard DeLong, A Profile of Mathematical Logic, 71–81.